Turkey has launched a series of aggressive airstrikes against Kurdish militants but has yet to turn its firepower on Islamic State in Syria as expected, increasing concerns in Washington about the Ankara government’s intentions.

Three weeks ago, the countries announced a breakthrough agreement to allow the U.S. to use bases in Turkey to launch airstrikes against Islamic State, and the Ankara government said it would join in the bombings. On Wednesday, the Pentagon said U.S. jets now based at Incirlik Air Base in southern...